Question: $\int (-9 x^2 +9 x +2)\,dx=$ $+C$
We can use the sum rule and the constant multiple rule for indefinite integrals: $\begin{aligned} &\int [f(x)+g(x)]dx=\int f(x)\,dx+\int g(x)\,dx \\\\\\ &\int k\cdot f(x)= k\cdot\int f(x)\,dx \end{aligned}$ Using the sum and the constant multiple rules, we can rewrite our integral as follows: $\int (-9 x^2 +9 x +2)\,dx= -9\int x^2\,dx +9\int x\,dx +2\int 1\,dx$ Now we can find each indefinite integral using the reverse power rule: $\int x^n\,dx=\dfrac{x^{n+1}}{n+1}+C$ Note: we can only use the reverse power rule because $n \neq -1$. $\begin{aligned} &\phantom{=}\int (-9 x^2 +9 x +2)\,dx \\\\ &= -9\int x^2\,dx +9\int x\,dx +2\int 1\,dx \\\\ &=-9 \dfrac{x^3}{3} +9\dfrac{x^2}{2} +2\dfrac{x^1}{1}+C \\\\ &=-3 x^3 +\dfrac{9}{2} x^2 +2 x+C \end{aligned}$ In conclusion, $\int (-9 x^2 +9 x +2)\,dx=-3 x^3 +\dfrac{9}{2} x^2 +2 x+C$